MoL-2008-12:
Yang, Fan
(2008)
*Intuitionistic Subframe Formulas, NNIL-Formulas and n-universal Models.*
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## Abstract

In this thesis, we investigate intuitionistic subframe formulas and

NNIL- formulas by using the technique of n-universal

models. Intuitionistic sub- frame formulas axiomatize subframe logics

which are intermediate logics characterized by a class of frames

closed under subframes. Zakharyaschev in- troduced the subframe

formulas by using [^,->]-formulas, which contain only ^ and -> as

connectives. It then follows that subframe logics are axiomatized by

[^,->]-formulas.

NNIL-formulas are the formulas that have No Nesting Of Implications to

the left. Visser, de Jongh, van Benthem and Renardel de Lavalette

proved that NNIL-formulas are exactly the formulas preserved under

taking submodels. The topic of this thesis was inspired by

N. Bezhanishvili who used the insight that NNIL-formulas are then

preserved under subframes as well to introduce subframe formulas in

the NNIL-form. It was proved that NNIL-formulas are su±cient to

axiomatize subframe logics.

This thesis is set up in a way to be able to connect the results on

subframe formulas defined by [^,->]-formulas and NNIL-formulas by

using n-universal models as a uniform method. Our original intention

to throw new light on subframe logics by the use of NNIL-formulas was

barely realized, but we do provide new insights on the NNIL-formulas

themselves.

Chapter 2 gives a background on intuitionistic propositional logic and

its Kripke, algebraic and topological semantics. In Chapter 3, we

discuss n-universal models U(n) of IPC by giving proofs of known

theorems in a uniform manner including a direct and very perspicuous

proof of the fact that the n-universal model of IPC is isomorphic to

the upper part of the n-Henkin model. This then also gives a method

for a new proof (Theorem 3.4.9) of Jankov's theorem on KC. In Chapter

4, we summarize classic and recent results on subframe logics and

subframe formulas. In Chapter 5, we investigate properties of the

[^,->]-fragment of IPC consisting of [^,->]- formulas only. This

chapter is based on the results in Diego, de Bruijn and Hendriks. We

redefined the exact model defined by using the n-universal models of

IPC and give a uniform treatment of known results.

In Chapter 6, we give an algorithm to translate every NNIL-formula to

a [^,->]-formula in such a way that they are equivalent on frames. We

study subsimulations between models and construct representative

models for equivalence classes of rooted generated submodels of U(n)

induced by two-way subsimulations. We construct finite n-universal

models U(n)^NNIL for NNIL-formulas with n variables by the

representative models and prove the related properties. As a

consequence, the theorem that formulas pre- served under

subsimulations are equivalent to NNIL-formulas becomes a natural

corollary of the properties of U(n)^NNIL. Finally, we obtain the

subframe logics axiomatized by two-variable NNIL-formulas by observ-

ing the structure of U(2)NNIL. Although it is not yet clear how to

general- ize the result for the model U(2)NNIL and the subframe logics

axiomatized by NNIL(p,q)-formulas to the models U(n)^NNIL for any n 2

->, this result clearly suggests that the U(n)^NNIL models are a good

tool for future work on subframe logics.

Item Type: | Report |
---|---|

Report Nr: | MoL-2008-12 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2008 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/807 |

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